Was he or wasn't he? Did he or didn't he? Among the wilder claims in the more contentious areas of Shakespearean research is the notion that The Bard was Italian. Or if he wasn't, then he at least visited Italy, an idea based in part on the fact that 10 of his plays are wholly or partly set in the country. These include two – Othello and The Merchant of Venice – that take place mostly in Venice, home to Francesco da Mosto (right), presenter of the current BBC series Shakespeare in Italy.

Francesco – who is going to take me on a Venetian tour of his favourite Shakespearean "haunts" – is well placed to present such a series. He comes from a long line of Venetians – one his family's former homes, the Ca' da Mosto, is the oldest surviving palazzo on Venice's Grand Canal, and his ancestors include a cast of individuals who could easily pass as Shakespearean characters. "We didn't have doges," he says, "but merchants, scholars, jurists and, of course, some black sheep. An Elena da Mosto was Byron's mistress, Alvise da Mosto was a 15th-century navigator, and a certain Marchesina da Mosto was abbess of Venice's Convent of San Zaccaria and, possibly, a courtesan. And we know that two da Mosto brothers visited Southampton in 1319, when five Venetian galleys docked there," emphasising the far-reaching nature of Venetian trade, and suggesting just how easy it might have been for Shakespeare to travel to Venice.

Whether he came or not, he was certainly familiar with the city by repute. When, in The Merchant of Venice, for example, Launcelot gives directions saying, "Turn up on your right hand, at the next turning, but at the very next turning of all, on your left… turn of no hand but turn down indirectly…", anyone who has ever been lost in the narrow alleys will recognise this as Venice.

Francesco recalls his earliest Shakespearean memories. "I remember an English lesson at school, about 35 years ago, when the teacher was trying to teach us Shakespeare. I struggled even to pronounce his name… but the first play I saw aroused such strong and profound feelings in me. It was The Merchant of Venice, either at Teatro La Fenice or Teatro Goldoni."

La Fenice was destroyed by fire in 1996 and Francesco was part of the team appointed to create a successor for the theatre.

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"I worked with Aldo Rossi on the design. He sadly died during the process and it turned into a ridiculous and acrimonious battle between construction firms and politicians, resulting in what I think is a garish chocolate box.

"Before, it had the atmosphere of a fairy tale, now you see tacky trinkets on sale, even baseball caps with the 'Phoenix' logo. When did anyone going to the opera need one of those! I can only start breathing once the lights are dimmed and the performance begins."

We set off around Venice, where something of the atmosphere of Shakespeare's plays can be recaptured simply by strolling the streets and alleys. News and gossip are still heard "on the Rialto", amorous couples still drift along the canals in gondolas and Jews are still to be found in the distinctive Ghetto of northern Venice. This was the first gated, guarded community for Jews and the word actually derives from the Venetian for foundry, geto.

"I imagine it was like a gilded prison for the Jews who lived there," says Francesco. "On the one hand they had safe homes, but on the other they were obliged to live by the many rules imposed by the Venetian State and were limited in the professions and trade they could take up. My family, among many, came to the Ghetto to borrow money."

The Museo Ebraico here, which holds a collection of ritual objects, operates guided tours of the area. Five of the original nine synagogues survive but only two are in use. There are several good kosher bakeries and restaurants, as well as many typical bars, but the Jewish community is much reduced.

Still, there are plenty signs of life. "My youngest son's favourite place, the Laboratorio Blu children's bookshop, is just off the Campo del Ghetto Nuovo," says Francesco. "And I also enjoy the Galleria Ikona, which specialises in modern photography."

Nearby is one of Francesco's favourite Venetian places, the Campo dei Mori – evoking Shakespeare's "Moor", Othello. It's home to several statues of mori, including a famous turbaned man with a broken nose – but he isn't Othello, simply one of the three brothers Mastelli who were Venetian silk traders from Morea in the Peleponnese.

Another contender for Shakespeare's tragic hero, the Doge Cristoforo Moro, sometimes dubbed "the true Othello", can be visited where he lies with his wife in a beautiful tomb in front of the altar in the nearby church of San Giobbe. And there's the Palazzo Contarini-Fasan, known as "Desdemona's House", on the Grand Canal, though its connections with the tragedy, sadly, are fanciful.

Who, then, was the real Othello? "Curiously enough," says Francesco, "I can throw some light on this. Research carried out by my grandfather Andrea, director of the Venice Archives in the Thirties, showed him to be a southern Italian, known as 'Il Moro' or 'Capitano Moro', possibly because of his dark skin. His name was Francesco da Sessa and in 1544 he was sent in chains from Cyprus to Venice, where he was imprisoned by order of the Council of Ten. In the end he was banished from the Republic for 10 years. Considering the relative lightness of the sentence, the crime may well have been one of passion."

Walking south, we come to one of Francesco's favourite spots for an evening drink, or ombra, the dark little Cantina do Mori (impossible to escape this Moorish connection) near the Rialto, which pre-dates Shakespeare – customers have been served wine and snacks here since 1462. "It's easy to feel at home in Do Mori," says Francesco, "as in all the bacari (traditional bars) in Venice: the habit of chatting or reading, with some cichetti (snacks) and a glass of wine never dies.

"And in the same narrow street not far from here," he continues, "are two places that offer traditional Venetian cuisine and an enjoyable atmosphere. Alla Madonna is an old favourite, and a few steps away is Osteria al Diavolo e l'Aquasanta, a cosy, unpretentious family-run restaurant."

The Rialto is the heart of the "modern" and "Shakespearean" city – but is there any difference? Almost any campo would serve as a backdrop for a Shakespearean street scene, any palazzo interior a setting for witty dialogue. Is any other city so little changed since Shakespeare's day?

The Rialto bridge may then have been made of wood but the Erberia and Pesceria markets, dating from 1097, continue to supply the city with fresh produce. "Given the canals and surrounding lagoon, a view from any of the bell towers must look the same as what Shakespeare would have seen – had he ever come here. But that is a separate story! Shakespeare or not, though, every place here is a jewel, a stage set, so lose yourself and let Venice guide you into its labyrinth."

Francesco da Mosto presents 'Shakespeare in Italy' on BBC Two on Thursday at 9pm

Getting there

Packages

Kirker Holidays (020 7593 2283; kirkerholidays.com) offers a three-night package to Venice from £546 per person, including return flight, breakfast, transfer by shared private water taxi, guaranteed entrance tickets for the Doges’ Palace or the Guggenheim or Accademia galleries, a copy of the Kirker Guide Notes and the services of a Kirker concierge.

Il Palladio Hotel and Spa £££ The celebrated Bauer Hotel has diversified at Giudecca with this perfect restoration of a convent originally designed by the 16th-century architect Andrea Palladio. Newly opened, the very luxurious five-star offers a wide range of spa treatments, three magnificent gardens and a complimentary shuttle to San Marco (041 520 7022; palladiohotelspa.com; doubles from €300/£245).

Palazzo Sant’ Angelo £££ Comfortable rooms, good bathrooms and a magnificent position right on the Grand Canal, with vaporetto stop outside. Be sure to get a room with a view and you will be entertained by the mix of gondolas and everyday traffic which ply this amazing waterway (041 241 1452; palazzosantangelo.com; doubles from €350/£285).

The best restaurants

Cantina do Mori £ A favourite haunt of Francesco’s; an atmospheric old osteria for wine by the glass and tapas-like cichetti (Calle del Do Mori 429; 041 522 5401).

Alla Madonna ££ Popular with locals such as Francesco as well as visitors, this lively and long-established trattoria, a stone’s throw from the Rialto markets, does wonderful fish cooked in authentic Venetian style such as delicious sarde in soar (sardines pickled with raisins and pine nuts) and fritto di pesce (mixed fried fish). As in most Venetian restaurants it is not worth buying expensive wine – the house white is a perfect accompaniment. The service is friendly and professional and the mood informal (Calle della Madonna 594; 041 522 3824).

Antica Trattoria alla Maddelena ££ Francesco suggests a visit to the islands of the northern lagoon where in this restaurant on the island of Mazzorbo, over the bridge from Burano, you can enjoy dishes such as “wild duck and artichokes” (Fondamenta Santa Caterina Mazzorbo 7; 041 730151).